I'm providing an overview of some classical results about automorphic forms in one section of a paper I'm working on, and I've realized I don't have a good reference for the following.

Let $k$ denote a number field, $\mathbb{A}$ the ring of $k$-adeles, and $G$ the $\mathbb{A}$-points of a reductive algebraic group defined over $k$ (possibly with additional restrictions if they are needed). Then $G = \prod'G_\nu$ is a restricted product of groups $G_\nu$ defined over the $\nu$-adic completion $k_\nu$ of $k$ for each place $\nu$ of $k$. Let $G_\infty = \prod_{\nu \mid \infty} G_\nu$, $\mathfrak{g} = \text{Lie}(G_\infty)$, let $K$ denote a fixed maximal compact subgroup of $G$, and let $G_k$ denote the $k$-points of the same reductive group. By an automorphic form I mean a smooth function $f: G \to \mathbb{C}$ that is left $G_k$-invariant, $K$-finite, $\mathcal{Z}$-finite for $\mathcal{Z}$ the center of the complexified universal enveloping algebra of $\mathfrak{g}$, and of moderate growth. (This is, e.g., the same definition found in Bump.)

Denote the right regular action of $G_\infty$ on these forms by $r$ and by $\mathtt{d}r$ the derived action of the complexified universal enveloping algebra $\mathcal{U}(\mathfrak{g}_\mathbb{C})$. Say that an automorphic form is of uniform moderate growth if, for some norm $\|\cdot\|$ on $G$, there exists $d \geq 0$ such that for all $u \in \mathcal{U}(\mathfrak{g}_\mathbb{C})$ there exists $C_u > 0$ such that

$$ |\mathtt{d}r(u)f(g)| \leq C_u\|g\|^d, \qquad\forall g \in G. $$

I believe I have seen a proof before that all automorphic forms (as defined above) are of uniform moderate growth. Is this true or are more restrictions on $G$ needed? Also, I believe I have seen authors mention that this was originally proved (presumably for $G$ a real reductive group) by Harish-Chandra. Can someone point me to this reference?

Thanks for your help!


1 Answer 1


At long last, I found the original result! This is a consequence of Theorem 1 (on p. 18) in

Harish-Chandra, Discrete series for semisimple groups. II Explicit determination of the characters, Acta Math. 116, 1966, 1-111.

For the sake of the next weary Ph.D. candidate to come along searching for this reference, what follows is the content of the aforementioned theorem and how it connects to moderate growth.

First, fix a semisimple (or reductive) Lie group $G$ (over $\mathbb{R}$ or $\mathbb{C}$) and let $K$ denote a maximal compact subgroup of $G$, $\mathfrak{g} = \operatorname{Lie}(G)_\mathbb{C}$ the complexification of the Lie algebra of $G$, $\mathcal{U}(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$, and $\mathfrak{z}$ the center of $\mathcal{U}(\mathfrak{g})$. Let $V$ denote a finite dimensional complex vector space. Theorem 1 states the following:

Theorem 1. Fix a $C^\infty$-function $f: G \to V$ that is $K$- and $\mathfrak{z}$-finite. For each neighborhood $U$ of $1$, let $J(U)$ denote the subspace of $C_c^\infty(G)$ consisting of all functions $\alpha$ supported in $U$ and invariant under the bi-regular representation of $K$ on $C_c^\infty(G)$ (i.e., such that $\alpha(kgk^{-1}) = \alpha(g)$ for all $k \in K$ and all $g \in G$). For each neighborhood $U$ of $1$, there exists $\alpha \in J(U)$ such that $f = f * \alpha$.

Although it might only seem tangentially related at first glance, Theorem 1 has the following corollary (not stated explicitly in Harish-Chandra's paper):

Corollary. Let $f$ be as in Theorem 1. If $f$ is of moderate growth, then $f$ is of uniform moderate growth.

Sketch of Proof. For each $m \in \mathbb{Z}$, define an "extended" seminorm $\rho_m: C^\infty(G;V) \to [0,\infty]$ by

$$ \rho_m(F) = \sup_{g \in G} \|g\|^m \|F(g)\|, $$

where (1) $\|g\|$ denotes the value of some norm on $\operatorname{GL}_N(\mathbb{R})$ at the image of $g$ under a suitable embedding (or finite-to-one map in the general reductive case) and (2) $\|F(g)\|$ is the value of a norm $\|\cdot\|$ on $V$ at $F(g)$. To say that $f$ is of moderate growth is to say that $\rho_m(f) < \infty$ for some $m$. Likewise $f$ is of uniform moderate growth if and only if there exists $m$ such that $\rho_m(Df) < \infty$ for all differential operators $D$.

Fix an $m$ such that $\rho_m(f) < \infty$. Let $\alpha$ be as in Theorem 1 for some choice of neighborhood $U$. If $D$ is a differential operator, recall that its value on $f * \alpha$ is the function

$$ D(f * \alpha) = f * (D\alpha). $$

It is an easy exercise to show that $\rho_m(f * \beta) < \infty$ whenever $\rho_m(f) < \infty$ and $\beta \in C_c^\infty(G)$, so $D\alpha \in C_c^\infty(G)$ implies

$$ \rho_m(Df) = \rho_m(D(f * \alpha)) = \rho_m(f * (D\alpha)) < \infty, $$

establishing the Corollary.


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